Bruisers are tankier and typically slower but not to the level of tanks.
Think juggernauts when it comes to Bruiser. Darius, Morde, Skarner (Has speed, less damage) Olaf, Sion. Basically they're like tanks with the purpose to stick to targets, but tanks tend to pack CC and try and disable carries. Bruisers tend not to have too many disables, simply damage and beefiness. They aren't as tanky, building a couple damage items while tanks build fully tanky typically. They are similar to fighters in that they are designed to excel in just a constant fight. Without kiting or a strong burst, they're designed to win the DPS time war (Aside from late game ADCs, but the tankiness allows the bruiser to usually deplete the ADC's health bar first)
Fighters are more agile, higher damage but squishier targets. They have kits designed to get in, and simply duel/fight. They don't have burst kits, they have DPS kits. Typically they also come with some sort of good defensive ability. Think Riven, Yasuo, Fiora, Tryndamere. However, they are almost a reverse of bruisers. They typically build mostly damage and a couple tanky items.
So, what do fighters have that bruisers don't? Well that's easy. Let's compare two of them. Riven vs Darius. Riven is more agile, higher direct damage. She has many dashes, as well as a shield to add a bit of tankiness to her. Darius is slower. Does less damage, but he's tankier. Both from build paths and from the sustain he receives from Q. Riven more hard CC, Darius has slows and a slight knock up. Riven packs stun and knock up.
Let's compare Fiora to Morde. Again, Fiora is more mobile both if she lands her lunge and if she does not. If not, it's still a dash every some seconds. If she lands it, virtually no CD. Either way, it's more than Morde's got. She's squishier than Morde in obvious ways, and her damage is definitely higher. However, she does have innate tankiness in her parry ability. Morde is slow, lumbering, tankier, and has lower DPS but due to tankiness, has the same bruiser advantage of extended DPS fights. Fighters have an advantage in short DPS fights.
Now, these are polar opposite stereotypes of these archtypes. Let's try and deviate and still identify the differences.
Skarner and Tryndamere.
Skarner, unlike many of these bruisers, has speed in both his shield and his passive. He's also got a stun and that ult plus a slow (I'm not saying bruisers have no CC and this is surprising, see Sion and Renekton, however it's a deviation from Morde and Darius that I used as examples.) So why is he still a bruiser? Well, unlike someone like Darius who is a bruiser that shares similarities with fighters, Skarner is a bruiser that shares similarities with tanks. Quite tanky, plenty of CC, decent way to initiate (Huge reason as to why he's not a tank, think Amumu-Sej-Naut). However, he still does damage due to his Q and the natural build path of sheen. He also usually builds a bit of AD, typically a sheen item and another AD item and then tanky, as is characteristic of a bruiser. Again, he has the advantage in the long DPS trade due to tankiness and constant stream of damage (Q-auto-sheen).
Tryndamere, unlike those fighters up there, typically builds full damage. Sometimes a Trynd may be cheeky and get a bit beefy, but overall, it's a 5 item powerhouse with boots attached. This leads one to believe "Well, why not assassin?" Because as often as all of us have been 3 hit by a fed Tryndamere due to 100% crit and shiv, that's not assassination. It's an extremely high stream of DPS. He has that characteristic mobility that fighters have to get in, he's got that innate tankiness due to his ult and he definitely has that advantage in the short DPS war. Trynd is an outlier due to the fact that he also wins long DPS wars usually due to that ult, but that's really the only characteristic he's got that's even somewhat bruiser. But that's how champion design works; not all champs were meant to be black and white. he sticks via dash and slow. Unlike many fighters, he's got no hard CC. But, this is due to the way his slow works and that he gets 5 seconds of guaranteed life.
So they exhibit properties considered foreign to their archtypes, but they still stick to the archtype they'd been given for the most part.
So even with the new archtypes Riot wants to instill, there are gray areas. I'd even argue there are more gray areas in the new system than the old. It makes the game look and sound flashier and more sophisticated though if, instead of saying "Support" we get to sip our tea cup and go "We need a Vanguard this game" or "A warden would really help us out."
Either way, it's almost all irrelevant. All these archtypes serve to do is create balance arguments. "But Yasuo is a skirmisher, not an assassin so it's not okay that he one shot me!" (Not complaining about Yasuo, I know he can't one shot. I'm making an example.) Classifying them doesn't change their kits, their playstyles, their anything. It's just something to make the game sound better and more appealing.
Also, disclaimer; Idk if I sound hostile. That's never the intention. So if you read this or any other comment in a hostile tone, that's not my intended vibe.
Does this explain the difference well or should I get some more examples?
makes you unable to do shit and you just lost the game.
since every other fighter builds defensive items and has the peel and utility of a tank.
shield+knock-up+surpress+projectile block, 
invurn+AD boost+crit chance boost+slows, and many more)
I figured she might come up. Now, here's the tricky part that bruiser and fighter have, and is probably the reason this new subclass system is coming in. Champions are never restricted to their design. If you build amumu full AP, he's a mage. Build him full AD, he's a shitty, shitty fighter. I say this because of Vi. Vi has extreme flexibility in her build. She can build to be a bruiser, a tank or a fighter. Build her warrior, Triforce/Maw and rest tank? Bruiser. Build her 3-4 damage items and tank items after? Fighter. Build her cinderhulk and full tank? Tank. I wouldn't call her full AD build assassin because she's not quick and agile like assasins. Even her ult is slow, her Q has a charge time, etc. Overall though, the effective way to play Vi is bruiser. She typically builds a couple damage items, then rest defense. She's got a little bit of mobility but nothing on the scale of Tryndamere or Riven due to charge up time. She has one form of CC, so she's not some Nautilus and she's just a beefy damage dealer.
